Combinatorics in Higher Solovay Models
Pith reviewed 2026-05-18 14:31 UTC · model grok-4.3
The pith
Assuming the consistency of ZFC with large cardinals, a model exists in which aleph_omega is a strong limit and L(P(aleph_omega)) satisfies the aleph_omega-perfect set property for all subsets of sequences, has no scale, fails SCH and AP,,
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Assuming the consistency of ZFC with appropriate large cardinal axioms we produce a model of ZFC where aleph_omega is a strong limit cardinal and the inner model L(P(aleph_omega)) satisfies the following properties: every set A subset (aleph_omega)^omega has the aleph_omega-PSP, there is no scale at aleph_omega, the singular cardinal hypothesis fails at aleph_omega, Shelah's approachability property fails at aleph_omega, and the tree property holds at aleph_omega+1. This provides the first example of a Solovay-type model at the level of the first singular cardinal and answers a question by Woodin on the relationship between the SCH and the AP at aleph_omega in ZF plus DC_aleph_omega.
What carries the argument
The forcing or inner-model construction that produces a model of ZFC in which aleph_omega is strong limit and L(P(aleph_omega)) satisfies the five listed combinatorial properties simultaneously.
If this is right
- Every subset of (aleph_omega)^omega has the aleph_omega-perfect set property inside the inner model.
- No scale exists at aleph_omega.
- The singular cardinal hypothesis fails at aleph_omega.
- Shelah's approachability property fails at aleph_omega.
- The tree property holds at aleph_omega+1.
Where Pith is reading between the lines
- The same style of construction may extend to produce analogous models at higher singular cardinals.
- The separation of SCH failure from AP failure inside a choiceless inner model suggests new independence results for these properties at singular cardinals.
- Minimal large-cardinal strength for these regularity properties at aleph_omega could be isolated by examining the exact assumptions used in the construction.
Load-bearing premise
The consistency of ZFC together with the appropriate large cardinal axioms must hold in order for the forcing or inner-model construction to arrange the five properties at aleph_omega.
What would settle it
An explicit construction of a scale at aleph_omega inside L(P(aleph_omega)) in every model where aleph_omega is a strong limit and SCH fails would show the claimed properties cannot coexist.
read the original abstract
We construe the singular-cardinal analogue of the classical Solovay model. Starting with large cardinal assumptions in the realm of supercompactness, we show that the our inner model captures a substantial portion of the combinatorics of $L(\mathcal{P}(\kappa))$ that are typically implied by Woodin's axiom $I_0$. Among other things, we show that in our higher Solovay model there are no $\kappa^+$-sequences of distinct members of $\mathcal{P}(\kappa)$ and that Shelah's approachability property $\AP_\kappa$ fails. We prove that every set in our inner model satisfies a singular analogue of the complete Ramsey property and that the partition relation $\kappa\xrightarrow[]{\mathrm{OD}} (\omega)^\omega_{V_\mu}$ holds for all $\mu<\kappa$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. Assuming the consistency of ZFC with appropriate large cardinal axioms, the paper constructs a model of ZFC in which ℵ_ω is a strong limit cardinal and L(𝒫(ℵ_ω)) satisfies five properties simultaneously: every A ⊆ (ℵ_ω)^ω has the ℵ_ω-PSP, there is no scale at ℵ_ω, SCH fails at ℵ_ω, AP fails at ℵ_ω, and TP holds at ℵ_ω+1. This is claimed to be the first Solovay-type model at the first singular cardinal and to answer Woodin's question on the relationship between SCH and AP in the context of ZF + DC_ℵ_ω.
Significance. If the construction succeeds, the result is significant for the study of singular cardinals. It extends Solovay-style models from inaccessible cardinals to ℵ_ω, showing that a combination of regularity-like properties (PSP, no scale, TP) can coexist with the failure of SCH and AP at the first singular cardinal. The simultaneous control of these properties in an inner model L(𝒫(ℵ_ω)) advances the understanding of what combinatorial theories are consistent at singulars and provides a concrete model answering a specific question of Woodin.
major comments (1)
- The central forcing or inner-model construction that arranges the five properties at once must be checked for preservation of the tree property at ℵ_ω+1 while forcing the failure of SCH and AP; without explicit verification that the iteration or collapse does not destroy TP, the claim that all five hold simultaneously remains load-bearing and requires a dedicated preservation lemma.
minor comments (2)
- The abstract refers to 'appropriate large cardinal axioms' without naming them; the introduction should list the specific assumptions (e.g., measurable or supercompact cardinals) used in the consistency proof.
- Notation for the ℵ_ω-PSP and the notion of 'scale at ℵ_ω' should be defined in a preliminary section before their use in the main argument.
Simulated Author's Rebuttal
We thank the referee for their detailed report and for highlighting the importance of explicitly verifying the preservation of the tree property. We address the major comment below and will incorporate the suggested clarification into a revised manuscript.
read point-by-point responses
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Referee: The central forcing or inner-model construction that arranges the five properties at once must be checked for preservation of the tree property at ℵ_ω+1 while forcing the failure of SCH and AP; without explicit verification that the iteration or collapse does not destroy TP, the claim that all five hold simultaneously remains load-bearing and requires a dedicated preservation lemma.
Authors: We agree that the simultaneous control of these properties benefits from a dedicated preservation argument. The construction proceeds from a supercompact cardinal via a carefully chosen iteration that first forces the failure of SCH and AP at ℵ_ω while preserving the tree property at ℵ_ω+1 from the ground model, and then takes the inner model L(𝒫(ℵ_ω)). In the revised version we will add a new lemma (provisionally Lemma 4.12) that isolates the preservation of TP(ℵ_ω+1) under the relevant forcing steps, citing the relevant facts from the literature on tree-property preservation at successors of singulars. This will make the argument self-contained without altering the overall construction. revision: yes
Circularity Check
No significant circularity; derivation is self-contained consistency result
full rationale
The paper presents a standard consistency result: from Con(ZFC + suitable large cardinals) it constructs (via forcing or inner-model analysis) a model of ZFC in which ℵ_ω is strong limit and L(𝒫(ℵ_ω)) satisfies the five listed properties simultaneously. The abstract and described claims contain no equations, fitted parameters, or self-definitional reductions that equate a target property to an input by construction. The large-cardinal assumptions are external background hypotheses drawn from prior literature rather than derived from the target model or from self-citation chains internal to this work. The central claim therefore remains independent of the conclusion and does not reduce to renaming, ansatz smuggling, or load-bearing self-reference.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Consistency of ZFC together with appropriate large cardinal axioms
Lean theorems connected to this paper
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IndisputableMonolith/Foundation/AbsoluteFloorClosure.leanreality_from_one_distinction unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
Main Theorem. Assume that ZFC is consistent with the existence of a supercompact cardinal and an inaccessible cardinal above it. Then, there is a generic extension V[G] where ... Every set A ⊆ (ℵ_ω)^ω has the ℵ_ω-PSP ... SCH fails at ℵ_ω ... AP fails at ℵ_ω
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IndisputableMonolith/Foundation/AlexanderDuality.leanalexander_duality_circle_linking unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
Lemma 2.38 (The Perfect Set Lemma) ... P, Q, R ... projections ... contains a copy of a κ-perfect set
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
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- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
discussion (0)
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